Why not use instrumental variable directly as a covariate in the regression? I know this is a silly question, as I know the theory of instrumental variables and two stage regression. Still, I never saw a clear answer to the following:


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*assume you have endogeneity due to unobserved variable correlated with one of the initial regressors. The typical way to correct for that is to find an instrumental variable correlated to the unobserved effect and to use a two-stage regression approach. 


Now my question is, why go through that trouble – why wouldn’t you just include the instrumental variable as a standard regressor in the initial estimation? 
 A: You can and people do. As @Alexis points out though, it doesn't give you a complete answer. 
Imagine you're interested in the effect of an endogenous variable $X$ on $Y$ and $Z$ is an instrument for $X$. When doing IV in econometrics:


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*The regression of $X$ on $Z$ is called the first stage regression.

*The regression of $Y$ on $Z$ is called the reduced form regression.


The reduced form regression on its own does not estimate the effect of $X$ on $Y$.
A: The point of instrumental variable regression is to provide an unbiased estimate of the causal effect of exposure $X$ on outcome $O$, when there is some unmeasured—possibly unmeasureable—variable $U$ confounding the relationship between $X$ and $O$. Here's a DAG of the simplest circumstance under which one would use instrumental variables estimation ($X$, $U$, and $Z$ can be sets of variables):

If an instrumental variable $Z$ causes $X$, has no effect on $O$ other than through $X$, there is no prior cause of both $Z$ and $O$, and the effect of $X$ on $O$ is homogeneous, then with a large enough sample $E[O|\hat{X}]$ where $\hat{X} = E[X|Z]$ can provide an unbiased estimate of the causal effect of $X$ on $O$.
In summary you do not care about the effect of $Z$ on $O$ (there is none except through $X$), and $E[O|\hat{X}] \ne E[O|X,Z]$, so simply including $Z$ in your model will not get you an instrumental variable estimate.
Final comment: The "...in the initial estimation?" closing of your question makes me want to clarify: one first estimates $\hat{X}$ (so $Z$ is indeed part of that estimation), and one uses $\hat{X}$ as a predictor in the second estimation (sans $Z$).
